Photoelastic investigation of large deflections by low-modulus photoelastic materials

Information is given to enable the experimental analysis of large deformations by application of photoelastic techniques. Promising photoelastic materials having low elastic modulus and high optical sensitivity were obtained for practical use at room temperature. The time and temperature dependence of the stress-fringe order and stress-strain relations of one of these viscoelastic materials is discussed. Photoelastic investigations of large deflected straight and circular beams are reported. Photoelastic technique seems to be a useful method for analysis of large deformations of models with complicated shapes.     

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

In the present paper, the isogeometric analysis (IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables (displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline (NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.     

变截面弹性直杆纵振动分析的小波--DQ法

在经典微分求积(DQ)法基础上, 根据多分辨分析理论, 以尺度函数为基础构造插值基函数, 形成了新的微分方程边值问题的求解方法------小波--DQ法, 并应用该方法分析了变截面弹性直杆的纵振动问题, 给出了其频率方程, 计算出了不同参数下固支--固支, 自由--自由楔形直杆和锥形直杆的固有频率,数值结果表明该方法是一个简单易行高精度的方法,该方法可以推广应用于其他力学问题的数值分析.     

Photoelastic investigation of crack-inclusion interaction

An experimental procedure is presented for determining the mode I stress-intensity factor of an edge crack with a nearby rigid elliptical inclusion in a finite plate loaded in uniform tension. The rigid inclusion was modeled by bonding two identical steel inclusions on to the faces of a thin plate of polycarbonate. Models were constructed with edge cracks and various inclusion geometries so that the effect of parameters such as inclusion shape, orientation, and cracktip position on the stress-intensity factors of the crack could be determined. Photoelasticity experiments were used for this investigation and the results were compared to the results of a similar theoretical analysis of the interaction between a crack and an inclusion in an infinite plate. A good correlation was found between the experimental and theoretical models indicating that the results may help provide a better understanding of the toughening mechanisms in materials such as short-fiber-reinforced composites and ceramics. This experimental method is relatively easy to use making it an attractive candidate to be applied to similar problems involving cracks and inhomogeneities.     

带凸缘弯曲梁应力集中区内表面裂纹应力强度因子的研究

用三维光弹性冻结应力实验技术与修正的多点超定法相结合研究了带凸缘弯曲梁应力集中区内表面裂纹的应力强度因子。分析了不同过渡圆弧的应力集中对两种表面裂纹(半圆形表面裂纹与前缘直线表面裂纹)的影响。用实验方法得到了应力强度因子放大系数的数值.结果表明,应力集中对浅裂纹的影响是更大的,是不可忽视的,但放大系数随表面裂纹的几何形状变化很小。这些对管节点的断裂力学评估提供了有价值的实验依据.     

An experimental and analytical treatment of matrix cracking in cross-ply laminates

The development of damage in cross-ply Hercules AS4/3502 graphite/epoxy laminates has been investigated. Specific endeavors were to identify the mechanisms for initiation and growth of matrix cracks and to determine the effect of matrix cracking on the stiffness loss in cross-ply laminates. Two types of matrix cracks were identified. These include both straight and curved cracks. The experimental study of matrix crack damage revealed that the curved cracks formed after the straight cracks and followed a repeatable pattern of location and orientation relative to the straight cracks. Therefore, it was postulated that the growth mechanism for curved cracks is driven by the stress state resulting from the formation of the straight cracks. This phenomenon was analytically investigated by a finite-element model of straight cracks in a cross-ply laminate. The finite-element results provide supporting evidence for the postulated growth mechanism. The experimental study also revealed that the number of curved cracks increased with the number of consecutive 90-deg plies. Finally, experimental results show as much as 10-percent degradation in axial stiffness due to matrix cracking in cross-ply graphite/epoxy laminates.     

钢筋混凝土T型梁在不同荷载(裂纹)下的试验模态分析及非线性外推

混凝土结构在大荷载作用下出现裂纹,其模态参数呈非线性规律.本文采用分段线性假设进行简化,即认为结构在不同级的荷载下对应有局部的线性特性.对钢筋混凝土T型简支梁在若干不同等级纯力加载并产生相应裂纹的情况下,分别进行试验模态分析,然后用非线性最小二乘拟合的方法,对所得各线性结果进行综合,以推求各种载荷作用(裂纹)下模态参数非线性变化规律.有关研究表明,将这种方法提供的结果用于桥梁承载力的动态法测定是合理可行的.     

Hydrostatic stress and crack opening displacement in three-point bend specimens with shallow cracks

A theoretical analysis is given for the slip-line field in a three-point bend specimen containing a sharp short crack. It is shown that the hydrostatic stresses are larger for deeper cracks. These results support the observation that the crack opening displacement at the initiation of a ductile tear is bigger for shallow cracks than deep ones. An analytical relationship for the ratio of plastic crack tip opening to crack mouth opening displacement is also established which is in agreement with experimental replica measurements. Application of the analysis to practical crack opening displacement measurements for shallow cracks is discussed.     

任意支承梁的固有频谱和模态的定性性质

确定了任意支承方式下欧拉梁横振动时其频谱和位移、转角、应变模态的一些重要特性，阐明了梁的位移模态的充分必要条件．     

Prevention of thermal bending of multilayered beams and plates

A method is described to prevent bending in multilayered beams and plates of different isotropic materials with uniform and nonuniform temperature distribution through the thickness. The method involved the addition of an extra layer to the multilayered beams or plates. With the proper selection of the thermoelastic properties, the added layer would eliminate the curvature produced prior to this addition. A complete analysis for the determination of the various thermoelastic parameters of the extra layer was made. In addition, to ensure that the multilayered beams and plates actually remained straight, a thermoelastic analysis was performed for the solution of thermal stresses and strains in the laminate. The results gave assurance to the straightness of the laminate since the calculated strains have the same value throughout the thickness. The solutions are valid for any given uniform temperature change and for any given nonlinear temperature distribution through the thickness of the multilayered beams and plates. Several numerical examples are presented that illustrate the application of the method for various temperature distributions. A simple experiment was conducted that showed the validity of the analytical method. A brass strip was added to a bimetalic strip made of aluminum and steel at room temperature. The thickness of the brass strip was calculated from the theory to prevent bending. The trimetal strip was placed in a furnace and, as expected, it remained straight for varying temperatures.     

裂纹分析中的单节点二次边界元

提出一种新的边界单元:单节点二次元.利用这种单元,位移及其沿边界的切向导数在正规单元端点的连续条件自然得到满足.单节点二次元能很好地模拟角点处面力多值条件.特殊裂纹尖端单节点二次单元包括近裂纹尖端位移近似级数展开第二项.由于每个单元只有一个节点,计算程序大大简化.对直裂纹、圆弧裂纹和边裂纹进行了计算.数值计算结果表明,单节点二次边界单元计算精度高,收敛性好.     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

光纤传感器测量FRP加固结构的损伤

A distributed fiber optic sensor is developed for condition monitoring of civil infrastructure sys-tems. The fiber optic sensor is especially useful in applications involving structures strengthened by fiberreinforced polymer (FRP) composites. The sensor principles are simple and therefore, practical for detec-tion of cracks, debonding and deformation measurements. Structural monitoring capability of the sensor     

Results of studying creep and long-term strength of metals at the Institute of Mechanics at the Lomonosov Moscow State University (To Yu. N. Rabotnov’s Anniversary)

Basic results of experimental and theoretical research of creep processes and long-term strength of metals obtained by researchers of the Institute of Mechanics at the Lomonosov Moscow State University are presented. These results further develop and refine the kinetic theory of creep and long-duration strength proposed by Yu. N. Rabotnov. Some problems arising in formulating various types of kinetic equations and describing experimental data for materials that can be considered as statically homogeneous materials (in studying the process of deformation and rupture of such materials, there is no need to study the evolution of individual cracks) are considered. The main specific features of metal creep models at constant and variable stresses, in uniaxial and complex stress states, and with allowance for one or two damage parameters are described. Criterial and kinetic approaches used to determine long-term strength under conditions of a complex stress state are considered. Methods of modeling the metal behavior in an aggressive medium are described. A possibility of using these models for solving engineering problems is demonstrated.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams

This paper deals with the detection of open cracks in beam structures that undergo transverse vibrations. The investigation is aimed at detecting the location of open cracks in damaged beams by minimizing measurement data and baseline information of the structure. The study is carried out by using the continuous wavelet transform (CWT). The application of this recent, but advanced, mathematical tool is initially presented through a theoretical background, which is believed to be valuable for bridging the gap between the CWT and previous existing techniques. It is shown how the possibility to efficiently identify localized damages by CWT comes up from the intrinsic capability of the wavelets to collect several mathematical tools in only one mathematical aspect: derivatives, convolution and appropriate smoothing of data are translated into the CWT. Simulations show how the redundancy of the CWT in the functional space is able to efficiently identify locations of open cracks in the presence of noisy or clean data. Indeed, the possibility to approach the problem by using different families of wavelets, for several available scales, allows a successful application of the characteristic microscopy of the wavelets. The technique may be promisingly applied to discrete vibrational data.     

Magnetoelastic buckling: Theory vs. experiment

Existing theoretical values and experimental values of magnetoelastic critical buckling fields for ferromagnetic beams and plates differ by a factor of two. The usual theoretical treatment assumes that the plate (a) is immersed in an infinite uniform field, (b) is infinitely wide and long, and (c) is made of a linearly magnetic material. We present an experimental investigation, combined with a simple analysis, which determines specific criteria which must be satisfied by experimental programs in order to validate the three assumptions. It is shown that experimental arrangements which validate (a) and (c) are feasible, but not (b). Hence, there is need for improved theoretical treatments which take into account the finite size of the specimens.     

Nonlinear vibrations of a sandwich piezo-beam system under piezoelectric actuation

The paper describes the use of active structures technology for deformation and nonlinear free vibrations control of a simply supported curved beam with upper and lower surface-bonded piezoelectric layers, when the curvature is a result of the electric field application. Each of the active layers behaves as a single actuator, but simultaneously the whole system may be treated as a piezoelectric composite bender. Controlled application of the voltage across piezoelectric layers leads to elongation of one layer and to shortening of another one, which results in the beam deflection. Both the Euler–Bernoulli and von Karman moderately large deformation theories are the basis for derivation of the nonlinear equations of motion. Approximate analytical solutions are found by using the Lindstedt–Poincaré method which belongs to perturbation techniques. The method makes possible to decompose the governing equations into a pair of differential equations for the static deflection and a set of differential equations for the transversal vibration of the beam. The static response of the system under the electric field is investigated initially. Then, the free vibrations of such deformed sandwich beams are studied to prove that statically pre-stressed beams have higher natural frequencies in regard to the straight ones and that this effect is stronger for the lower eigenfrequencies. The numerical analysis provides also a spectrum of the amplitude-dependent nonlinear frequencies and mode shapes for different geometrical configurations. It is demonstrated that the amplitude–frequency relation, which is of the hardening type for straight beams, may change from hard to soft for deformed beams, as it happens for the symmetric vibration modes. The hardening-type nonlinear behaviour is exhibited for the antisymmetric vibration modes, independently from the system stiffness and dimensions.

     

Fracture behaviors of piezoelectric materials

Theoretical analyses and experimental observations of the failure and fracture behaviors of piezoelectric materials are presented. The theoretical analyses are based on the Stroh formalism. A strip dielectric breakdown model is proposed to estimate the effect of electrical non-linearity on the piezoelectric fracture of electrically insulated cracks. The reviewed experiments include the indentation fracture test, the bending test on smooth samples, the fracture test on pre-notched or pre-cracked samples, the environment-assisted fracture test, etc. For electrically insulated cracks, the experimental results show a complicated fracture behavior under combined electrical and mechanical loading. Fracture data are greatly scattered when a static electric field is applied. For electrically conducting cracks, the experimental results demonstrate that static electric fields can fracture poled and depoled lead zirconate titanate (PZT) ceramics. A charge-free zone model is introduced to understand the failure behavior of conducting cracks in the depoled lead zirconate titanate ceramics under electrical and/or mechanical loading. These theoretical and experimental results indicate that fracture mechanics concepts are useful in the study of the failure behaviors of piezoelectric materials.     

Viscoelastic analysis of residual stress in quenched thermosetting resin beams

The residual stress generated by the molding process of thermosetting resins exerts serious influences upon their mechanical properties. This residual stress is generally classified by two groups: one produced by shrinkage in the curing reaction of monomers, the other produced by the nonuniformity of the temperature distribution in the cooling process. This paper is concerned with the theoretical and experimental analysis of the generation of residual stress of the latter type, using examples of rectangular beams of thermosetting resins quenched on both the upper and lower surfaces. First, a viscoelastic model is applied to make a qualitative prediction of the residual stress in quenched beams. Second, using linear-viscoelastic theory, fundamental equations are derived for the residual stress in a viscoelastic rectangular beam, where an unsteady and nonuniform temperature distribution is assumed in the direction of depth. The theoretical values of the residual stress in rectangular beams are calculated under various quenching conditions for two resins having different viscoelastic characteristics, i.e., epoxy and unsaturated polyester. The theoretical residual-stress distributions agree fairly well with the residual stress measured experimentally at every quenching condition for both resins. The qualitative prediction that the residual stress in quenched beams is compressive in the vicinity of the upper and lower surfaces and is tensile in the inner parts is confirmed. The relaxation modulus of epoxy resin changes more greatly with time and temperature than that of unsaturated polyester resin. The theoretical and experimental analysis shows that the residual stress for the former resin is larger than that for the latter. Therefore, it is concluded that the generation of residual stress is more significant where the relaxation modulus of resin changes greatly with time and temperature.